3D equlibrium averaged description and consistency check

1 Background Modelling of 3D equilibrium plasma configurations is a challenging task. The existence of magnetic islands and stochastic magnetic field regions makes a direct [1, 2] modelling time consuming, not very robust and flexible, and hardly useful for systematic equilibrium optimization and stability analysis. More tractable conventional model for 3D MHD equilibrium and stability studies is based on the nested magnetic surface approximation as in standard 3D equilibrium code VMEC [3]. However the code convergence is sensitive to the choice of harmonic set for flux surface representation and significantly deteriorates with increasing resolution. For equilibrium and stability studies based on the averaging methods a lot of numerical codes have been developed and good agreement for both equilibrium and stability in planar-axis stellarators was obtained [4]. The approach to 2D description of MHD equilibrium and stability proposed in [5] is more general. The key idea is to introduce Riemannian space R, in which reference 3D equilibrium is symmetric. The first step in such interpretation was carried out in [6, 7], where it was shown that for arbitrary 3D equilibrium (with nested magnetic surfaces at least) one can construct some formal 2D metric tensor and obtain 2D Grad-Shafranov type equation. The equation was obtained by averaging exact 3D equation. In fact, it is the exact zero 2D moment of equilibrium equation, like Kruskal-Kulsrud equation is the exact zero 1D moment. 2 Scalar equations for 3D MHD equilibria description By assuming the magnetic surfaces a(r) = const exist the ideal MHD equilibrium problem